# d_Helmholtz_3D accuracy

## d_Helmholtz_3D accuracy

What is the accuracy of d_Helmholtz_3D? I would like to know what the residual error is after solving the equation.

Thanks.

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Hi,d_Helmholtz_3D is the direct solver of matrix correspond of 7-point grid Helmholtz equation. So its provide accuracy based on floating operations.With best regards,Alexander Kalinkin

since it is double precision, does it mean the accuracy is 1E-16?

Dear Alexander,

I checked the accuracy of d_Helmholtz_3D and it was much much larger than 1E-16. Since the solver uses a standard seven-point discretization, I verified the accuracy by the following code. (I have a uniform mesh for my problem.)

for ( k=1; kGetTuple(  i    +  j   *NX +  k   *NX*NY);

phi_im1jk = Phi->GetTuple( (i-1) +  j   *NX +  k   *NX*NY);

phi_ip1jk = Phi->GetTuple( (i+1) +  j   *NX +  k   *NX*NY);

phi_ijm1k = Phi->GetTuple(  i    + (j-1)*NX +  k   *NX*NY);

phi_ijp1k = Phi->GetTuple(  i    + (j+1)*NX +  k   *NX*NY);

phi_ijkm1 = Phi->GetTuple(  i    +  j   *NX + (k-1)*NX*NY);

phi_ijkp1 = Phi->GetTuple(  i    +  j   *NX + (k+1)*NX*NY);
X_i   = X->GetTuple( i   );

X_im1 = X->GetTuple( i-1 );
Y_j   = Y->GetTuple( j   );

Y_jm1 = Y->GetTuple( j-1 );
Z_k   = Z->GetTuple( k   );

Z_km1 = Z->GetTuple( k-1 );
rhs= f->GetTuple( i + j*NX+ k*NX*NY);
res = ( phi_im1jk + phi_ip1jk - 2*phi_ijk)/pow( X_i - X_im1, 2 ) +

( phi_ijm1k + phi_ijp1k - 2*phi_ijk)/pow( Y_j - Y_jm1, 2 ) +

( phi_ijkm1 + phi_ijkp1 - 2*phi_ijk)/pow( Z_k - Z_km1, 2 ) +

rhs;
}

}

}

When I print res, the residual is about 1E-1. Is there something that I have to be careful when using the function? I need to have an accuracy about 1E-16. Please advise.

Thanks,

Hi Ahmad,To verify it I need to have full example with rhs and boundary condition. Could you provide this example to me by e'mail or by private answer?With best regards,Alexander Kalinkin

Alexander,

I found a bug in my code which misled me to the see large residuals. I fixed it and the error now is about 1E-15.

Thanks,

Hi Alexander,

I have another question. The solver is for uniform mesh. Does this mean that it has to have dx=dy=dz? Or we can have dx!=dy!=dz (constant dx, dy, dz everywhere in the domain)?

Thanks,

Hi Ahmad,The uniform mesh mean that all mesh steps are equals in one direction, but mesh sizes for different dimension could be differ. For example hx=0.2, hy=0.5, hz=0.1.With best regards,Alexander Kalinkin

I am writing a journal paper in which I have used d_Helmholtz. Regarding the 7-point grid Helmholtz equation, can I have the name of the method by which the system is solved?

Thanks,

Hi Ahmad,The main information could be in paper prepared by us a several years ago so feel free to use it.With best regards,Alexander Kalinkin

Thanks. You helped me a lot.
Best,

Hi Alexander,

I didn't find the method by which the library solves the system. Is it gradient bi-conjugate,
multigrid, overrelaxation, or Fourier? I would appreciate it.

Thanks,